Non \(\Sigma _ n\) axiomatizable almost strongly minimal theories.

*(English)*Zbl 0698.03021A theory is almost strongly minimal if in every of its models, every element is in the algebraic closure of a strongly minimal set. Hodges and MacIntyre conjectured in 1970 that there is a natural number n for which every \(\aleph_ 0\)-categorical and almost strongly minimal theory is \(\Sigma_ n\)-axiomatizable. G. Ahlbrandt and J. T. Baldwin proved recently [Arch. Math. Logic 27, No.1, 1-4 (1988; Zbl 0637.03022)] that every theory satisfying the above assumption of the Hodges-MacIntyre conjecture is \(\Sigma_ n\)-axiomatizable for some n.

In the paper under review, it is shown that for each n there is an \(\aleph_ 0\)-categorical almost strongly minimal theory which is not \(\Sigma_ n\)-axiomatizable. More precisely, the author proves the following: Theorem. For every n there is an almost strongly minimal \(\aleph_ 0\)-categorical theory T with models M, N of T such that N is a \(\Sigma_ n\)-elementary but not \(\Sigma_{n+1}\)-elementary extension of M. The rest follows from a theorem of C. C. Chang [J. Symb. Logic 32, 61-74 (1967; Zbl 0161.005)].

In the paper under review, it is shown that for each n there is an \(\aleph_ 0\)-categorical almost strongly minimal theory which is not \(\Sigma_ n\)-axiomatizable. More precisely, the author proves the following: Theorem. For every n there is an almost strongly minimal \(\aleph_ 0\)-categorical theory T with models M, N of T such that N is a \(\Sigma_ n\)-elementary but not \(\Sigma_{n+1}\)-elementary extension of M. The rest follows from a theorem of C. C. Chang [J. Symb. Logic 32, 61-74 (1967; Zbl 0161.005)].

Reviewer: P.Štěpánek

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##### References:

[1] | Omitting types of prenex formulas 32 pp 61– (1967) · Zbl 0161.00503 |

[2] | Archive for Mathematical Logic |

[3] | DOI: 10.1016/0168-0072(86)90037-0 · Zbl 0592.03018 · doi:10.1016/0168-0072(86)90037-0 |

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